Small time approximation in Wright-Fisher diffusion

Wright-Fisher model has been widely used to represent random variation in allele frequency over time, due to its simple form even though, a closed analytical form for the allele frequency has not been constructed. However, variation in allele frequency allows to represent selection in the evolutionary process. In this work, we present two alternatives of parametric approximating functions: Asymptotic expansion (AE) and Gaussian approximation (GaussA), obtained by means of probabilistic tools and useful for statistical inference, to estimate the allele frequency density for a small $t/2N$ in the interval $[0,1]$. The proposed densities satisfactorily capture the problem of fixation at 0 or 1, unlike the commonly used methods. While Asymptotic Expansion defines a suitable density for the distribution allele frequency (DAF), Gaussian Approximation describes a range of validity for the Gaussian distribution. Through a simulation study and using an adaptive method for density estimation, the proposed densities are compared with the beta and Gaussian distribution with, their corresponding parameters.